Q. 94

Question

Use algebra, limit rules, and the continuity of $e^{x}$ to prove that every exponential function of the form $f (x) = A e^{kx}$ is continuous everywhere.

Step-by-Step Solution

Verified

Answer

$T h e f u n c t i o n f (x) = A e^{kc} i s c o n t i n u o u s e v e r y w h e r e$ .

1Step 1. Given Information:

The strategy is to prove using algebra, limit rules, and the continuity of $e^{x}$ that every exponential function of the form $f (x) = A e^{k x}$ is continuous everywhere.

2Step 2. Prove:

$T a k e l i m i t i n t h e e x p r e s s i o n e^{x} a s : \underset{x \to c}{l i m} e^{x} = e^{c} T h e r e f o r e, \underset{x \to c}{l i m} A e^{k x} = \underset{x \to c}{l i m} A \underset{x \to c}{l i m} e^{k x} = A \lim_{x \to c} e^{k x} = A e^{k (c)} = A e^{k c} H e n c e, t h e f u n c t i o n f (x) = A e^{kc} i s c o n t i n u o u s e v e r y w h e r e .$

Q. 92

Q. 95

Other exercises in this chapter

Q. 91

Prove the constant multiple rule for limits:limx→cf(x)=L and k∈ℝ, then limx→ckf(x)=kL

View solution

Q. 92

Prove the difference rule for limits by applying the sum and constant multiple rules for limits.

View solution

Q. 95

Use algebra, limit rules, and the continuity of ex to prove that every exponential function of the form f(x)=Abx is continuous everywhere.

View solution

Q. 96

Use algebra, limit rules, and the continuity of ln x on (0,∞) to prove that every logarithmic function of the form f(x)=logbx is continuous on (

View solution